1074 MR. R. W. DYSON ON THE POTENTIAL OF AN ANCHOFl RING. 
Let OC = c : CA = a : CB = 6. 
Let X and z be the coordinates of any point in this section of the ring referred to 
CA and CB as axes. 
The potential of the ring at any external point ct', z , is 
(c — x) dx dz d(j) 
{(c — xy + — 2cj' (c — x) cos<j) + {z' — zf} ' 
■ («). 
_ f'- 
J J J 0 A 
c d(f) 
} 0 \/(vj'" + cos 4* + 
(42). 
Writing D for djdc, and D' for djdz, and taking the double integral over the area 
of the ellipse, we obtain for the potential at an external point, 
V. 
— ^TTUO <2^2.4= 2 t ^ 27476 
(« 2 D 2 +' r-B'J 
4! 
+ • 
f 
J n 
c d(j) 
J o\/+ d — 2cm' COS 
(43). 
§ 25. When the cross-section is circular, the formula may be simplified. Calling 
Therefore 
dcf) 
V/3 
dd 
+ 
d? ' 
dz'\ 
d'^ ' 
+ 
~d7\ 
' d^ 
+ 
dc^ 
d7^ 
\/(m'^ + — 2cm' COS 4> + 
"A + 'Zl + 1 A = 0 . 
dd^ ^ dz'^- ^ c dc 
dl 
do 
'i\> 
dl 
_ 
^ dc 
do 
d [cV- 
+ 
dd 
I oi 
1 
hi 
II 
dz'\ 
- i 
fcP 
+ 
~ dc ' 
ydd^ 
d 
~ ~ dc ' 
d /Idl 
do \c dc 
j i ca 
dz'y \c dc 
c dc\c dc 
- + 1C*) 
So that when the cross-section is circular, we arrive at the formula given in 
my former paper, p. 59. 
Vo = 
M 
ttc 
1 j. (1 ly 
8c dx 192 \ c dcj 
-•h: 
C d<j) 
-f- c" — 2ctij^ cos 
(44). 
